Understanding the Frustum of a Cone

In the previous chapter, "Melting and Reshaping Solids," we looked at what happens when we completely melt down a shape and cast it into something entirely new. But what if we don't destroy the shape completely? What if we just take a sharp knife and cleanly slice the top off?

If you take a standard right circular cone—like a birthday party hat or a traffic cone—and slice off its pointed top with a cut that is perfectly parallel to the ground, you are left with two pieces. The top piece is just a smaller cone. But the bottom piece, the one left sitting on the ground, is a completely new shape.

This bottom piece is called the frustum of a cone.

You have seen this shape every single day of your life. A standard steel bucket in your bathroom is an inverted frustum. The glass you drink water from is often a frustum. A lampshade is a frustum. Even the paper cup you get at a coffee shop is a frustum. Because it is so common in the real world, examiners love testing it.

In this chapter, we are going to tear this shape apart. We will learn its anatomy, understand where its measurements come from, and master the formulas for its volume and surface areas. Most importantly, we will learn how to memorize these complex-looking formulas by connecting them to the simple cone formulas you already know.

What Exactly is a Frustum?

Let us start by getting a precise picture of what we are dealing with.

Imagine a solid wooden cone resting on a table. You take a saw and make a straight, horizontal cut through the wood, perfectly parallel to the table. You remove the top pointed section. The heavy wooden base that remains on the table is the frustum.

Because a frustum is created by slicing a cone parallel to its circular base, the top cut you just made is also a perfect circle. This means a frustum has two circular faces, one at the bottom and one at the top. Because the cone was tapering inward before you cut it, these two circles will always be of different sizes.

The Anatomy of a Frustum

To calculate anything about a frustum, we need to know its dimensions. A standard cone has three main measurements: radius (r), vertical height (h), and slant height (l).

A frustum, however, has four main measurements, because it has two different radii. Let us define them clearly.

1. 1The Larger Radius (r₁): This is the radius of the larger circular base. If you are looking at a bucket sitting normally on the ground, the larger radius is at the open top.
2. 2The Smaller Radius (r₂): This is the radius of the smaller circular base. In our bucket example, this is the flat bottom resting on the floor.
3. 3The Vertical Height (h): This is the straight-line, perpendicular distance between the two parallel circular bases. If you drop a plumb line straight down from the center of the top circle to the center of the bottom circle, that distance is h.
4. 4The Slant Height (l): This is the shortest distance along the curved outer surface of the frustum, from the rim of the top circle to the rim of the bottom circle.

Note on notation: Standard textbooks usually use r₁ for the larger radius and r₂ for the smaller radius (so r₁ > r₂). Some books use R for the large radius and r for the small radius. Both mean the exact same thing. We will use r₁ and r₂ here to match the most common official curriculum standards.

Finding the Slant Height (l)

In a standard cone, the vertical height (h), the radius (r), and the slant height (l) form a perfect right-angled triangle. Because of this, we can always find the slant height using Pythagoras' theorem: l = √(h² + r²).

Does a frustum have a similar rule? Yes, but we have to adjust it because of the two different radii.

Imagine slicing the frustum straight down the middle, from top to bottom, and looking at the flat cross-section. The shape you would see is an isosceles trapezoid. The top horizontal line is the diameter of the small circle (2r₂). The bottom horizontal line is the diameter of the large circle (2r₁). The vertical distance between them is h, and the slanted sides are l.

If you draw a straight vertical line down from the edge of the smaller top circle to the larger bottom circle, you create a right-angled triangle on the outer edge of the frustum.

• The vertical side of this triangle is the height, h.
• The hypotenuse of this triangle is the slant height, l.
• The base of this triangle is the difference between the two radii: (r₁ - r₂).

Applying Pythagoras' theorem to this small triangle gives us the master formula for the slant height of a frustum:

l = √(h² + (r₁ - r₂)²)

The Volume of a Frustum

If you fill a frustum-shaped glass with water, how much water does it hold? This is its volume (or capacity).

The formula for the volume of a frustum looks intimidating at first glance, but it has a beautiful, logical structure.

Volume of a Frustum = 1⁄3 π h (r₁² + r₂² + r₁ r₂)

Let us break down why this makes sense. You already know the formula for the volume of a standard cone: V = 1⁄3 π h r².

Notice that the frustum formula starts exactly the same way: 1⁄3 π h. The only difference is that the single r² from the cone formula has been replaced by a longer bracket: (r₁² + r₂² + r₁ r₂).

Why? Because a frustum doesn't have just one radius. It has a large radius at one end and a small radius at the other. The term (r₁² + r₂² + r₁ r₂) acts as a sort of "geometric average" of the areas of the two bases. It accounts for the large base, the small base, and the smooth transition between them.

Curved Surface Area (CSA)

The Curved Surface Area (CSA) is the area of the slanted, outer wall of the frustum. If you were painting the outside of a lampshade, the CSA is the area you would paint. It does not include the flat circular top or the flat circular bottom.

Again, let us look at the standard cone formula for CSA: π r l.

For a frustum, we have two radii, so we replace the single r with the sum of the two radii (r₁ + r₂).

CSA of a Frustum = π l (r₁ + r₂)

It is that simple. Just remember that you must use the slant height (l) for surface area, never the vertical height (h). If a problem gives you h but asks for surface area, your very first step must be to calculate l using the Pythagoras formula we learned earlier.

Total Surface Area (TSA) and the "Open/Closed" Trap

The Total Surface Area (TSA) is the total amount of material needed to build the entire 3D shape, including its flat bases.

For a solid, completely closed frustum (like a heavy wooden paperweight), the TSA is the curved surface area PLUS the area of the large bottom circle PLUS the area of the small top circle.

TSA of a solid Frustum = CSA + Area of large base + Area of small base TSA = π l (r₁ + r₂) + π r₁² + π r₂²

The Real-World Trap

Examiners love to test whether you are blindly memorizing formulas or actually visualizing the shape. They do this by giving you objects that are open.

If a question asks for the total surface area of a bucket or a drinking glass, you must stop and think. A bucket is open at the top. It only has a curved surface and a bottom base. It does not have a top base.

If you blindly apply the full TSA formula to a bucket, you will get the wrong answer because you will be adding a lid that doesn't exist.

For an open bucket (where r₁ is the open top and r₂ is the closed bottom): Surface Area of a Bucket = CSA + Area of bottom base Surface Area = π l (r₁ + r₂) + π r₂²

Always read the question carefully to determine if the frustum is closed at both ends, open at one end, or open at both ends (like a pipe or a lampshade, which would just be the CSA).

The Master Cheat Sheet: Cone vs. Frustum

To ensure you never forget these formulas, memorize them as pairs. Notice how every frustum formula is just the cone formula with the radius part expanded to accommodate two different radii.

• Slant Height (l)
• Cone: l = √(h² + r²)
• Frustum: l = √(h² + (r₁ - r₂)²)
• The change: r becomes the difference between the radii.

• Volume (V)
• Cone: V = 1⁄3 π h (r²)
• Frustum: V = 1⁄3 π h (r₁² + r₂² + r₁ r₂)
• The change: r² becomes the sum of the squares plus their product.

• Curved Surface Area (CSA)
• Cone: CSA = π l (r)
• Frustum: CSA = π l (r₁ + r₂)
• The change: r becomes the sum of the radii.

• Total Surface Area (TSA)
• Cone: TSA = π l r + π r²
• Frustum: TSA = π l (r₁ + r₂) + π r₁² + π r₂²
• The change: Add the areas of both circular bases instead of just one.

How to Tackle a Frustum Problem

When you face a frustum question in an exam, follow this exact sequence to avoid careless errors.

Worked Example 1: The Solid Block

Let us put everything together with a straightforward calculation.

The Setup: A solid wooden block is in the shape of a frustum of a cone. Its vertical height is 4 cm. The radius of its larger circular base is 10 cm, and the radius of its smaller circular base is 7 cm. Find its slant height, volume, curved surface area, and total surface area. (Leave answers in terms of π).

Step 1: Identify the variables

• h = 4 cm
• r₁ = 10 cm
• r₂ = 7 cm

Step 2: Find the slant height (l) We use the slant height formula: l = √(h² + (r₁ - r₂)²) l = √(4² + (10 - 7)²) l = √(4² + 3²) l = √(16 + 9) l = √(25) l = 5 cm.

Step 3: Calculate the Volume V = 1⁄3 π h (r₁² + r₂² + r₁ r₂) V = 1⁄3 π (4) (10² + 7² + (10 × 7)) V = 1⁄3 π (4) (100 + 49 + 70) V = 1⁄3 π (4) (219) Notice that 219 is divisible by 3 (219 ÷ 3 = 73). V = π × 4 × 73 V = 292π cm³.

Step 4: Calculate the Curved Surface Area (CSA) CSA = π l (r₁ + r₂) CSA = π (5) (10 + 7) CSA = π (5) (17) CSA = 85π cm².

Step 5: Calculate the Total Surface Area (TSA) Because the problem states it is a "solid wooden block," it is closed at both ends. We must add the areas of both circular bases to the CSA. Area of large base = π r₁² = π (10²) = 100π Area of small base = π r₂² = π (7²) = 49π TSA = CSA + Large Base + Small Base TSA = 85π + 100π + 49π TSA = 234π cm².

Worked Example 2: The Open Bucket

Now let us look at a scenario where visualizing the object is critical.

The Setup: A metal bucket is in the shape of a frustum of a cone. Its vertical height is 15 cm. The radius of its open top is 14 cm, and the radius of its flat bottom is 6 cm. How much sheet metal is required to make this bucket? (Leave the answer in terms of π).

Step 1: Identify the variables

• h = 15 cm
• r₁ = 14 cm (top)
• r₂ = 6 cm (bottom)

Step 2: Find the slant height (l) l = √(h² + (r₁ - r₂)²) l = √(15² + (14 - 6)²) l = √(15² + 8²) l = √(225 + 64) l = √(289) l = 17 cm.

Step 3: Determine the required surface area The question asks for the amount of sheet metal required. A bucket has a curved side and a flat bottom, but no top lid. Therefore, we need the CSA plus the area of the smaller bottom base only.

Area of sheet metal = CSA + Area of bottom base Area = π l (r₁ + r₂) + π r₂² Area = π (17) (14 + 6) + π (6²) Area = π (17) (20) + 36π Area = 340π + 36π Area = 376π cm².

If you had blindly used the full TSA formula, you would have added the area of the 14 cm top opening (196π), resulting in a completely wrong answer. Always visualize the object!

Quick Revision

• A frustum is created by slicing the top off a cone parallel to its base.
• It has four key measurements: vertical height (h), slant height (l), larger radius (r₁), and smaller radius (r₂).
• Slant Height: l = √(h² + (r₁ - r₂)²)
• Volume: V = 1⁄3 π h (r₁² + r₂² + r₁ r₂)
• Curved Surface Area: CSA = π l (r₁ + r₂)
• Total Surface Area (Solid): TSA = π l (r₁ + r₂) + π r₁² + π r₂²
• Exam Trap: Always check if the real-world object (like a bucket or glass) is open at one end. If it is, do not use the full TSA formula; only add the area of the closed base to the CSA.
• Memorization Trick: Frustum formulas are just cone formulas where the single radius r has been expanded to account for r₁ and r₂.